curve #455

y² = x³ + x² − 51118531933701314117226670804090500x + 4356025409211819471328533670264706621678809638000000

a-invariants: [0, 1, 0, -51118531933701314117226670804090500, 4356025409211819471328533670264706621678809638000000]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 9046965486159523716544575196047665163733977446752919090057367280896972900840
discriminant (Δ): 351813999866196822692970241266591817067249227481505285438859517294302644182141346113538486221794212000000
Faltings height: 18.8456
naive height: 251.3720
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 29, 31, 761, 299420740859957, 659019010527062691669524340330116973975278291527
regulator: 58858858571422723844.2868667811299677054077138364645781730950536605996620943155
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:55:06 UTC
last updated: 2026-08-30 15:55:06 UTC

Witness: 20 independent points log in to add more points →

(154424326982378250, -12025713959336719806459750)
(146371335632552100, 3111307220644996728209400)
(303200116114304325/4, -242224255380339872638751025/8)
(-257453653841194650, -21261145790228525147543850)
(146866893716796960, 4038235095728219235791760)
(3780175372430021859/25, -1143429954303110039273887452/125)
(263131439432161125/4, 285989641761929977008057375/8)
(172134654493731168, -25635186573774056717214816)
(150339389334688890, -8298023083945805269319190)
(167995846115145300, -22574425259649525038758800)
(435551997540570645/4, 71942998795782752438535735/8)
(176848961643645375, -29100095945333582653307250)
(112802037489144255, 5007772069381269154454670)
(114573464037418638, -1792485426397757275412046)
(229956561305985675, -69000660633652654168735200)
(145617025316912310, -81058880209224691936890)
(301464304001768874, -127839382609714743765008250)
(156303627273346350, 13588224304357200687584850)
(85528871188559778, 24689569290552663476676606)
(450666550807329045/4, -41443411904600605502053815/8)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -41/6. The 3 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 20.

last edited by Bhavik Mehta at 2026-08-30 15:55:06 UTC · history